Can anyone explain how to approach such problems? I tried finding fitting lemmas, but I can't say I can put the pieces together..
Consider the predicates $P_1,P_2,Q_1,Q_2 : R$ $\rightarrow \{0,1\}$ such that $P_1 \vee P_2 $ , $Q_1 \vee Q_2 $ are semi-decidable and $P_2 = \overline{Q_2}$. What can be said about $P_1 \vee Q_1 $?
On
Absolutely nothing. (I think there's probably a typo in what you wrote . . .)
To see this, set $P_2=True$, $Q_2=False$, and $Q_1=False$. Then $P_1\vee P_2=True$ and $Q_1\vee Q_2=False$, so each are semi-decidable. Meanwhile, $P_1\vee Q_1$ is just $P_1$. But there are no constraints on what $P_1$ can be.
Let us look at number-theoretic predicates. Suppose that $P_1$ is an arbitrarily badly undecidable subset of the natural numbers. Let $P_2$ be the set of natural numbers, and let $Q_1$ be say the empty set.
Then $P_1\lor P_2$ is semi-decidable, indeed decidable, as is $Q_1\lor Q_2$, but $P_1\lor Q_1$ is not even semi-decidable.